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Properties

Label 16.16.127...496.1
Degree $16$
Signature $[16, 0]$
Discriminant $1.273\times 10^{31}$
Root discriminant \(87.91\)
Ramified primes $2,47$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4^2.(C_2\times C_4)$ (as 16T362)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 + 772*x^12 - 5792*x^10 + 22430*x^8 - 45872*x^6 + 47348*x^4 - 21056*x^2 + 2209)
 
Copy content gp:K = bnfinit(y^16 - 48*y^14 + 772*y^12 - 5792*y^10 + 22430*y^8 - 45872*y^6 + 47348*y^4 - 21056*y^2 + 2209, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 48*x^14 + 772*x^12 - 5792*x^10 + 22430*x^8 - 45872*x^6 + 47348*x^4 - 21056*x^2 + 2209);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 48*x^14 + 772*x^12 - 5792*x^10 + 22430*x^8 - 45872*x^6 + 47348*x^4 - 21056*x^2 + 2209)
 

\( x^{16} - 48x^{14} + 772x^{12} - 5792x^{10} + 22430x^{8} - 45872x^{6} + 47348x^{4} - 21056x^{2} + 2209 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[16, 0]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(12725851295326073899065850986496\) \(\medspace = 2^{70}\cdot 47^{6}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(87.91\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{305/64}47^{1/2}\approx 186.48547112066305$
Ramified primes:   \(2\), \(47\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{4}a^{4}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}+\frac{1}{4}a$, $\frac{1}{29547985756}a^{14}-\frac{653429497}{7386996439}a^{12}-\frac{6402736613}{29547985756}a^{10}-\frac{606688775}{7386996439}a^{8}+\frac{6274428947}{29547985756}a^{6}-\frac{73897785}{157170137}a^{4}+\frac{14759712605}{29547985756}a^{2}+\frac{20175521}{157170137}$, $\frac{1}{29547985756}a^{15}-\frac{653429497}{7386996439}a^{13}+\frac{492129913}{14773992878}a^{11}-\frac{1}{4}a^{10}+\frac{4960241339}{29547985756}a^{9}-\frac{1}{4}a^{8}+\frac{6274428947}{29547985756}a^{7}-\frac{73897785}{157170137}a^{5}+\frac{3686358083}{14773992878}a^{3}+\frac{1}{4}a^{2}-\frac{76468053}{628680548}a+\frac{1}{4}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $15$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{6914}{8038081}a^{14}-\frac{312508}{8038081}a^{12}+\frac{4453068}{8038081}a^{10}-\frac{54220673}{16076162}a^{8}+\frac{71496502}{8038081}a^{6}-\frac{1356179}{171023}a^{4}-\frac{10096680}{8038081}a^{2}+\frac{938539}{342046}$, $\frac{54876391}{14773992878}a^{14}-\frac{5137482701}{29547985756}a^{12}+\frac{39305655133}{14773992878}a^{10}-\frac{542038676239}{29547985756}a^{8}+\frac{907977937283}{14773992878}a^{6}-\frac{61030361993}{628680548}a^{4}+\frac{878725072813}{14773992878}a^{2}-\frac{3188351007}{628680548}$, $\frac{18896216}{7386996439}a^{14}-\frac{3533564509}{29547985756}a^{12}+\frac{13474004508}{7386996439}a^{10}-\frac{368992552275}{29547985756}a^{8}+\frac{305218873337}{7386996439}a^{6}-\frac{40594060949}{628680548}a^{4}+\frac{306723623701}{7386996439}a^{2}-\frac{4831420007}{628680548}$, $\frac{6052505}{29547985756}a^{14}-\frac{223980319}{29547985756}a^{12}+\frac{1626929019}{29547985756}a^{10}+\frac{9153949467}{29547985756}a^{8}-\frac{137221229059}{29547985756}a^{6}+\frac{10089703383}{628680548}a^{4}-\frac{624073098441}{29547985756}a^{2}+\frac{6002530177}{628680548}$, $\frac{54558063}{7386996439}a^{14}-\frac{5037942347}{14773992878}a^{12}+\frac{75208452341}{14773992878}a^{10}-\frac{250993979086}{7386996439}a^{8}+\frac{1614961614633}{14773992878}a^{6}-\frac{25874207469}{157170137}a^{4}+\frac{711632088275}{7386996439}a^{2}-\frac{3422216537}{314340274}$, $\frac{54876391}{14773992878}a^{15}+\frac{56660765}{29547985756}a^{14}-\frac{5137482701}{29547985756}a^{13}-\frac{1305816021}{14773992878}a^{12}+\frac{39305655133}{14773992878}a^{11}+\frac{38997056051}{29547985756}a^{10}-\frac{542038676239}{29547985756}a^{9}-\frac{131746049189}{14773992878}a^{8}+\frac{907977937283}{14773992878}a^{7}+\frac{882442969891}{29547985756}a^{6}-\frac{61030361993}{628680548}a^{5}-\frac{15573873227}{314340274}a^{4}+\frac{878725072813}{14773992878}a^{3}+\frac{1030254862041}{29547985756}a^{2}-\frac{3188351007}{628680548}a-\frac{1483474333}{314340274}$, $\frac{279902577}{14773992878}a^{14}-\frac{13004415053}{14773992878}a^{12}+\frac{196136807817}{14773992878}a^{10}-\frac{1322354176081}{14773992878}a^{8}+\frac{2141350563178}{7386996439}a^{6}-\frac{68717894478}{157170137}a^{4}+\frac{1889270683415}{7386996439}a^{2}-\frac{5168029160}{157170137}$, $\frac{92668823}{14773992878}a^{15}+\frac{211740487}{29547985756}a^{14}-\frac{4335523605}{14773992878}a^{13}-\frac{9849037753}{29547985756}a^{12}+\frac{66253664149}{14773992878}a^{11}+\frac{148946388695}{29547985756}a^{10}-\frac{455515614257}{14773992878}a^{9}-\frac{1009818773267}{29547985756}a^{8}+\frac{1518415683957}{14773992878}a^{7}+\frac{3305534611371}{29547985756}a^{6}-\frac{50812211471}{314340274}a^{5}-\frac{107957830367}{628680548}a^{4}+\frac{1492172320215}{14773992878}a^{3}+\frac{3050735395787}{29547985756}a^{2}-\frac{4324225781}{314340274}a-\frac{7919216905}{628680548}$, $\frac{189303577}{14773992878}a^{15}+\frac{491919363}{29547985756}a^{14}-\frac{8950897971}{14773992878}a^{13}-\frac{5743487425}{7386996439}a^{12}+\frac{279231100385}{29547985756}a^{11}+\frac{87550488785}{7386996439}a^{10}-\frac{1983005461319}{29547985756}a^{9}-\frac{2405770517047}{29547985756}a^{8}+\frac{1735268002024}{7386996439}a^{7}+\frac{8062187850261}{29547985756}a^{6}-\frac{62152832082}{157170137}a^{5}-\frac{68843463986}{157170137}a^{4}+\frac{7973613953129}{29547985756}a^{3}+\frac{2098587739121}{7386996439}a^{2}-\frac{22194646453}{628680548}a-\frac{20592226491}{628680548}$, $\frac{13984367}{29547985756}a^{15}+\frac{11431497}{29547985756}a^{14}-\frac{783681841}{29547985756}a^{13}-\frac{365097567}{29547985756}a^{12}+\frac{3960843202}{7386996439}a^{11}+\frac{131526290}{7386996439}a^{10}-\frac{37947263706}{7386996439}a^{9}+\frac{26065728925}{14773992878}a^{8}+\frac{730188043209}{29547985756}a^{7}-\frac{467366901857}{29547985756}a^{6}-\frac{35272779061}{628680548}a^{5}+\frac{30287465057}{628680548}a^{4}+\frac{345212893457}{7386996439}a^{3}-\frac{354491742377}{7386996439}a^{2}-\frac{352483836}{157170137}a+\frac{1567485013}{314340274}$, $\frac{12542250}{7386996439}a^{15}+\frac{22053587}{29547985756}a^{14}-\frac{2384785101}{29547985756}a^{13}-\frac{251211807}{7386996439}a^{12}+\frac{9381635016}{7386996439}a^{11}+\frac{14438176849}{29547985756}a^{10}-\frac{269334955301}{29547985756}a^{9}-\frac{43567440935}{14773992878}a^{8}+\frac{239513587999}{7386996439}a^{7}+\frac{225752741719}{29547985756}a^{6}-\frac{35608746945}{628680548}a^{5}-\frac{2541371259}{314340274}a^{4}+\frac{316002472621}{7386996439}a^{3}+\frac{109381141221}{29547985756}a^{2}-\frac{5299093593}{628680548}a-\frac{234908104}{157170137}$, $\frac{55304534}{7386996439}a^{14}-\frac{2619228183}{7386996439}a^{12}+\frac{81935961675}{14773992878}a^{10}-\frac{291872470289}{7386996439}a^{8}+\frac{1023877528988}{7386996439}a^{6}-\frac{36563330247}{157170137}a^{4}+\frac{2301569142579}{14773992878}a^{2}-\frac{2709179325}{157170137}$, $\frac{222310803}{29547985756}a^{15}+\frac{281263525}{29547985756}a^{14}-\frac{10241442743}{29547985756}a^{13}-\frac{13007511617}{29547985756}a^{12}+\frac{151904668843}{29547985756}a^{11}+\frac{194192220361}{29547985756}a^{10}-\frac{996562071319}{29547985756}a^{9}-\frac{1283746133981}{29547985756}a^{8}+\frac{3080937795997}{29547985756}a^{7}+\frac{4019861236161}{29547985756}a^{6}-\frac{90710939651}{628680548}a^{5}-\frac{123108629407}{628680548}a^{4}+\frac{1985894966829}{29547985756}a^{3}+\frac{3261930268305}{29547985756}a^{2}+\frac{1778377701}{628680548}a-\frac{8702127603}{628680548}$, $\frac{101318933}{14773992878}a^{15}+\frac{2765409}{157170137}a^{14}-\frac{9158293333}{29547985756}a^{13}-\frac{253809571}{314340274}a^{12}+\frac{65401411169}{14773992878}a^{11}+\frac{3737724023}{314340274}a^{10}-\frac{808815055427}{29547985756}a^{9}-\frac{24289161973}{314340274}a^{8}+\frac{574273671080}{7386996439}a^{7}+\frac{37228459277}{157170137}a^{6}-\frac{62617046935}{628680548}a^{5}-\frac{104498532461}{314340274}a^{4}+\frac{394914915719}{7386996439}a^{3}+\frac{57227732205}{314340274}a^{2}-\frac{4613649289}{628680548}a-\frac{5972093545}{314340274}$, $\frac{970484803}{29547985756}a^{15}-\frac{331381931}{14773992878}a^{14}-\frac{45465789379}{29547985756}a^{13}+\frac{30522329209}{29547985756}a^{12}+\frac{174390351054}{7386996439}a^{11}-\frac{452253241721}{29547985756}a^{10}-\frac{1212349919229}{7386996439}a^{9}+\frac{1479231408111}{14773992878}a^{8}+\frac{16633789008545}{29547985756}a^{7}-\frac{2271204811662}{7386996439}a^{6}-\frac{598633393163}{628680548}a^{5}+\frac{262411852219}{628680548}a^{4}+\frac{5346441671555}{7386996439}a^{3}-\frac{5156074156505}{29547985756}a^{2}-\frac{30225847963}{157170137}a-\frac{3492079126}{157170137}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 39170296162.0 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 39170296162.0 \cdot 1}{2\cdot\sqrt{12725851295326073899065850986496}}\cr\approx \mathstrut & 0.359801830189 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 48*x^14 + 772*x^12 - 5792*x^10 + 22430*x^8 - 45872*x^6 + 47348*x^4 - 21056*x^2 + 2209) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 48*x^14 + 772*x^12 - 5792*x^10 + 22430*x^8 - 45872*x^6 + 47348*x^4 - 21056*x^2 + 2209, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 48*x^14 + 772*x^12 - 5792*x^10 + 22430*x^8 - 45872*x^6 + 47348*x^4 - 21056*x^2 + 2209); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 48*x^14 + 772*x^12 - 5792*x^10 + 22430*x^8 - 45872*x^6 + 47348*x^4 - 21056*x^2 + 2209); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_4^2.(C_2\times C_4)$ (as 16T362):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 128
The 14 conjugacy class representatives for $C_4^2.(C_2\times C_4)$
Character table for $C_4^2.(C_2\times C_4)$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), 8.8.592973922304.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 32 siblings: data not computed
Arithmetically equivalent siblings: data not computed
Minimal sibling: 16.16.12725851295326073899065850986496.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.8.0.1}{8} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ R ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.1.16.70e1.3402$x^{16} + 8 x^{14} + 8 x^{10} + 4 x^{8} + 16 x^{7} + 56 x^{4} + 2$$16$$1$$70$16T362$$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}, \frac{21}{4}]$$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{47}$$x + 42$$1$$1$$0$Trivial$$[\ ]$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.1$x^{2} + 47$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.2.1.0a1.1$x^{2} + 45 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.1.2.1a1.2$x^{2} + 235$$2$$1$$1$$C_2$$$[\ ]_{2}$$
47.2.2.2a1.2$x^{4} + 90 x^{3} + 2035 x^{2} + 450 x + 72$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)